Republican Susan Collins and a dozen other senators from both parties were busy last weekend hashing out what looked like a budget compromise, a rare glimmer of hope amid a bitter impasse that shut the government and threatened a debt default.

But top Senate Democrats hated the emerging deal, which included spending limits and changes to the health-care law, and so did the White House. After the White House chief of staff phoned lawmakers and urged them to back away, five Democrats working with Ms. Collins issued a...